cp's OEIS Frontend

A332743 Number of non-unimodal compositions of n covering an initial interval of positive integers.

%I A332743 #7 Feb 16 2025 08:33:59
%S A332743 0,0,0,0,0,1,5,14,35,83,193,417,890,1847,3809,7805,15833,32028,64513,
%T A332743 129671,260155,521775,1044982,2092692,4188168,8381434,16767650,
%U A332743 33544423,67098683,134213022,268443023,536912014,1073846768,2147720476,4295440133,8590833907
%N A332743 Number of non-unimodal compositions of n covering an initial interval of positive integers.
%C A332743 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332743 A composition of n is a finite sequence of positive integers summing to n.
%H A332743 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332743 For n > 0, a(n) = A107429(n) - A227038(n).
%e A332743 The a(5) = 1 through a(7) = 14 compositions:
%e A332743   (212)  (213)   (1213)
%e A332743          (312)   (1312)
%e A332743          (1212)  (2113)
%e A332743          (2112)  (2122)
%e A332743          (2121)  (2131)
%e A332743                  (2212)
%e A332743                  (3112)
%e A332743                  (3121)
%e A332743                  (11212)
%e A332743                  (12112)
%e A332743                  (12121)
%e A332743                  (21112)
%e A332743                  (21121)
%e A332743                  (21211)
%t A332743 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A332743 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332743 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&!unimodQ[#]&]],{n,0,10}]
%Y A332743 Not requiring non-unimodality gives A107429.
%Y A332743 Not requiring the covering condition gives A115981.
%Y A332743 The complement is counted by A227038.
%Y A332743 A version for partitions is A332579, with complement A332577.
%Y A332743 Unimodal compositions are A001523.
%Y A332743 Non-unimodal permutations are A059204.
%Y A332743 Non-unimodal normal sequences are A328509.
%Y A332743 Numbers whose unsorted prime signature is not unimodal are A332282.
%Y A332743 Cf. A007052, A072704, A072706, A332281, A332284, A332287, A332578, A332639, A332642, A332669, A332834, A332870.
%K A332743 nonn
%O A332743 0,7
%A A332743 _Gus Wiseman_, Mar 02 2020